What Is Equivalence Class in Relations and Functions Class 12: Definition & Examples

Definition of Equivalence Class in Class 12 Mathematics

An equivalence class is a subset formed by elements of a set that are equivalent to each other under a given equivalence relation.

Formally, if $R$ is an equivalence relation on a set $A$, then for any element $a \in A$, the equivalence class of $a$ is:

$$[a] = \{ x \in A : (a, x) \in R \}$$

This means $[a]$ contains all elements related to $a$ by $R$. Equivalence classes help group elements sharing a common property defined by the relation.

In Class 12 NCERT, understanding equivalence classes is key to grasping how relations partition sets into disjoint groups.

Properties of Equivalence Relations and Their Classes

An equivalence relation $R$ on a set $A$ must satisfy three properties:

• Reflexive: Every element is related to itself, i.e., $(a, a) \in R$ for all $a \in A$.
• Symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
• Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

Because of these properties, equivalence classes have these features:

• Every element $a$ belongs to its own equivalence class $[a]$.
• Equivalence classes are either identical or disjoint.
• The set of all equivalence classes forms a partition of $A$.

These properties ensure the equivalence relation neatly groups elements.

How Equivalence Classes Partition a Set

Equivalence classes divide a set $A$ into non-overlapping subsets such that every element of $A$ belongs to exactly one equivalence class. This is called a partition of the set.

Key points:

• The union of all equivalence classes equals the entire set $A$.
• No two distinct equivalence classes share any element.

For example, consider the set $A = \{1, 2, 3, 4, 5, 6\}$ with relation $R$ defined as "having the same remainder when divided by 3".

• $[1] = \{1, 4\}$ (remainder 1)
• $[2] = \{2, 5\}$ (remainder 2)
• $[3] = \{3, 6\}$ (remainder 0)

These classes are disjoint and cover all elements of $A$.

This partitioning helps simplify problems by focusing on groups rather than individual elements.

Examples of Equivalence Classes in Class 12 NCERT Problems

Let's solve a common example to understand equivalence classes better.

Example 1:

Set $A = \{1, 2, 3, 4, 5, 6\}$, relation $R$ defined as $aRb$ if $a - b$ is divisible by 3.

• Check if $R$ is an equivalence relation:
• Reflexive: $a - a = 0$ divisible by 3 ✔
• Symmetric: If $a - b$ divisible by 3, then $b - a$ divisible by 3 ✔
• Transitive: If $a - b$ and $b - c$ divisible by 3, then $a - c$ divisible by 3 ✔

• Equivalence classes:
• $[1] = \{1, 4\}$
• $[2] = \{2, 5\}$
• $[3] = \{3, 6\}$

Example 2:

Set $B = \{a, b, c, d\}$, relation $S$ defined by $aSb$ if $a$ and $b$ have the same first letter.

• Since all elements are single letters, $S$ relates each element only to itself.
• Equivalence classes are singletons: $[a] = \{a\}$, $[b] = \{b\}$, etc.

These examples illustrate how equivalence classes group elements sharing a relation.

Difference Between Equivalence Relation and Equivalence Class

Understanding the distinction between an equivalence relation and an equivalence class is crucial:

In simple terms, the equivalence relation is the rule, and equivalence classes are the groups formed by this rule.

Applications of Equivalence Classes in Class 12 Mathematics

Equivalence classes have several important applications in Class 12 Mathematics and beyond:

• Simplifying Relations: They help break down complex relations into manageable groups.
• Partitioning Sets: Equivalence classes partition sets into disjoint subsets, useful in counting and classification problems.
• Functions and Mappings: Understanding equivalence classes aids in defining quotient sets and functions on these sets.
• Modular Arithmetic: Equivalence classes under modulo relations form the basis of modular arithmetic, a key topic in number theory.

By mastering equivalence classes, students gain deeper insight into relations and functions, which is essential for NCERT exam success.